Source-linked AI summary
Convex Relaxations and Linear Approximation for Optimal Power Flow in Multiphase Radial Networks
Lingwen Gan, Steven H. Low
TL;DR
Multiphase radial distribution networks require tractable and stable methods for OPF and power-flow computation. The paper proposes two SDP relaxations and a linear power-flow approximation, proving equivalent exactness of the relaxations and reporting numerical exactness and close voltage estimates in case studies.
Problem
The paper addresses how to obtain exact convex relaxations, compute them efficiently, and improve numerical stability for OPF in multiphase radial networks.
Method
The paper proposes BIM-SDP and BFM-SDP for OPF, plus LPF for estimating power flows and voltages.
Results
BFM-SDP is numerically exact in the evaluated networks, while LPF obtains voltages within 0.0016 per unit of their true values.
Takeaways & Limitations
BIM-SDP improves computational efficiency, BFM-SDP improves numerical stability, and LPF provides accurate voltage estimates under small-loss, nearly balanced conditions.
Abstract
from arXiv · showhide
Distribution networks are usually multiphase and radial. To facilitate power flow computation and optimization, two semidefinite programming (SDP) relaxations of the optimal power flow problem and a linear approximation of the power flow are proposed. We prove that the first SDP relaxation is exact if and only if the second one is exact. Case studies show that the second SDP relaxation is numerically exact and that the linear approximation obtains voltages within 0.0016 per unit of their true values for the IEEE 13, 34, 37, 123-bus networks and a real-world 2065-bus network.
I. INTRODUCTION
The paper develops convex relaxations and a linear power-flow approximation for multiphase radial networks, addressing computational efficiency, numerical stability, and approximation accuracy. Its relaxations are evaluated on standard and real-world networks.
- The paper develops convex relaxations of OPF and a linear approximation of power flow for multiphase radial networks.
- The paper considers when convex relaxations are exact, how to compute them efficiently, and how to attain numerical stability.
- BIM-SDP exploits radial topology and reduces declared variables from O(n^2) in standard SDP to O(n).
- BFM-SDP avoids subtractions of voltages that are close in value, improving numerical stability while retaining BIM-SDP's computational efficiency.
- BFM-SDP is numerically exact on the IEEE 13, 34, 37, and 123-bus networks and a real-world 2065-bus network, whereas BIM-SDP is numerically exact only on the IEEE 13- and 37-bus networks.
- LPF is accurate when line loss is small and voltages are nearly balanced, achieving voltage errors within 0.0016 per unit on the evaluated networks.
B. Optimal Power Flow
OPF minimizes generation cost subject to physical and operational constraints on power injections and voltages. The formulation includes separable generation costs, device-specific injection sets, fixed substation voltage, and regulated branch-bus voltages.
- OPF determines power injections that minimize generation cost subject to physical and operational constraints.
- Generation cost is separable across buses, with each bus having its own cost function.
- Branch-bus power injections vary within externally specified sets that may be nonconvex or disconnected.
- The substation voltage is fixed and given, while branch-bus voltages are regulated within specified ranges.
- Voltage limits can represent a specified tolerance around nominal values, such as 0.95 to 1.05 per unit for a 5% range.
III. BIM SEMIDEFINITE PROGRAMMING
BIM-SDP reformulates nonconvex OPF for multiphase radial networks by shifting nonconvexity to rank constraints and removing them. It reduces the variable count relative to standard SDP and can recover global OPF optima when the relaxation is exact.
- BIM-SDP exploits radial network topology to reduce the computational complexity of standard SDP for nonconvex OPF.The nonconvexity is shifted from the original formulation to rank constraints, which are then removed.
- BIM-SDP may remain nonconvex when the objective functions Ci or constraint sets Si are nonconvex.The SDP relaxation is convex only under the stated convexity conditions, apart from the removed rank constraint.
- If an optimal BIM-SDP solution satisfies (5g), Algorithm 1 recovers a voltage solution that solves OPF globally.Theorem 1 states that the recovered voltage V makes (s, V) solve OPF.
- BIM-SDP is exact when every optimal BIM-SDP solution satisfies (5g).Under exactness, solving BIM-SDP yields a global optimum of OPF.
B. Comparison with a Standard SDP
The standard SDP uses a large positive-semidefinite matrix and does not exploit radial sparsity, while the branch flow model provides an equivalent formulation. BIM-SDP is computationally efficient but numerically unstable, motivating a separate stable relaxation.
- Comparison with a Standard SDP: Standard-SDP shifts OPF nonconvexity to rank ˜W = 1 and obtains a relaxation by removing that rank constraint.
- Comparison with a Standard SDP: BIM-SDP has O(n) variables, compared with O(n^2) variables in standard-SDP.
- Comparison with a Standard SDP: Standard-SDP leaves n^2 blocks of its matrix appearing only in the positive-semidefinite constraint, creating sparsity that BIM-SDP can exploit.The radial topology supports positive-sem definite matrix completion using a chordal extension.
- Comparison with a Standard SDP: BIM-SDP is not numerically stable, motivating a different SDP relaxation.The branch flow model avoids ill-conditioned voltage subtractions while retaining equivalent power-flow solutions.
B. Branch Flow Model Semidefinite Programming
The branch flow model introduces alternative variables and an equivalent OPF formulation to avoid ill-conditioned operations. Its SDP relaxation can recover global optima when its lifted constraint is satisfied, but convexity still depends on Ci and Si.
- A numerically stable SDP with computational efficiency similar to BIM-SDP is proposed for multiphase radial OPF.
- BFM-SDP may be nonconvex when Ci or Si are nonconvex.Removing (10g) yields an SDP relaxation only when the remaining objective and constraint components are convex.
- If an optimal BFM-SDP solution satisfies (10g), Algorithm 2 produces a global optimum of OPF’.Under this condition, the BFM-SDP solution also solves BFM-OPF.
C. Comparison with BIM-SDP
BFM-SDP reformulates BIM-SDP with alternative variables that avoid ill-conditioned voltage subtractions while retaining equivalent optimality and computational size.
- BFM-SDP avoids subtractions of closely valued voltages, improving numerical stability over BIM-SDP.
- BFM-SDP and BIM-SDP have the same number of variables and constraints, giving them similar computational efficiency.
- A bijective feasible-set map preserves objective values between BIM-SDP and BFM-SDP.
- The map also preserves correspondence between optimal solutions of the two relaxations.
- BIM-SDP is exact if and only if BFM-SDP is exact.
V. LINEAR APPROXIMATION
LPF is a linear multiphase power-flow approximation derived under small line losses and nearly balanced voltages, producing unique approximate flows and voltages from injections and substation voltage.
- LPF provides a linear approximation of power flows and voltages in multiphase radial networks.
- LPF assumes line losses are small compared with power flows.
- LPF approximates off-diagonal power-flow terms using diagonal terms when voltages are nearly balanced.
- Given non-substation injections and substation voltage, LPF uniquely determines substation injection, line flows, and downstream voltages.
- LPF generalizes the Simplified DistFlow Equations to multiphase networks without DC approximation assumptions of constant voltage magnitude, ignored reactive power, or zero resistance.
- Case studies evaluate LPF against forward-backward sweep results on five test networks.
A. BIM-SDP vs BFM-SDP
The study compares BIM-SDP and BFM-SDP across radial-network OPF cases and evaluates LPF accuracy, finding stronger numerical stability for BFM-SDP and close voltage estimates from LPF.
- BIM-SDP vs BFM-SDP: The experiments use BIM-SDP and BFM-SDP to compare running time, objective values, and proximity of solutions to rank one.
- BIM-SDP vs BFM-SDP: With 10% voltage flexibility, BFM-SDP is numerically exact for all test networks, whereas BIM-SDP is exact for only two.
- BIM-SDP vs BFM-SDP: BFM-SDP remains numerically exact for up to 2000-bus networks when OPF is feasible, while BIM-SDP encounters numerical difficulties from 34 buses.
- BIM-SDP vs BFM-SDP: The simulations include IEEE 13, 34, 37, and 123-bus networks plus a real-world 2065-bus network.
- Accuracy of LPF: Voltages estimated by LPF are within 0.0016 per unit of true values, and power flows are within 5.3% for all test networks.
- Conclusions: The paper presents BIM-SDP and BFM-SDP as convex OPF relaxations, with BIM-SDP targeting efficiency and BFM-SDP targeting numerical stability.
APPENDIX
The appendix establishes voltage-recovery and feasible-set results using induction and bidirectional mappings between BIM and BFM formulations.
- Algorithm 1 recovers voltages for all buses by iteratively expanding the visited set from the substation.
- The voltage-recovery property is proved by induction over algorithm iterations.
- Radial topology guarantees each newly reached bus has a unique parent already visited in the preceding iteration.
- Rank-one Hermitian matrices are represented through vectors α and β for bus and line voltage-related variables.
- The feasible-set equivalence proof constructs BIM variables from BFM variables and BFM variables from BIM variables.
- Both inclusion directions are completed to establish equivalence of the BIM and BFM feasible sets.
C. Proof of Lemma 2
The proof establishes Algorithm 2’s correctness by induction over visited network nodes and recovered voltages and currents, using radial structure and rank-one factorization.
- Conclusion: The induction concludes that Algorithm 2 computes (V, I) satisfying V0 = V ref and the stated network conditions.The proof then completes the correctness argument for Algorithm 2.
- Inductive recovery: Algorithm 2 recovers voltage V_i for every visited node and current I_ij for every induced edge after each iteration.After the final iteration, all network voltages and currents are recovered.
- Inductive recovery: The induction starts trivially at k = 0 and advances from iteration K to K + 1.The proof covers k = 0, 1, . . . , n.
- Inductive recovery: Radiality gives each newly visited node j a unique parent i that was visited in the preceding iteration.This unique-parent property reduces the induction step to one edge extension.
- Inductive recovery: The proof uses the Hermitian rank-one structure to represent relevant quantities through vectors α and β in C|Φij|.The cited factorization yields i = ααH, Sij = αβH, and ℓij = ββH.
D. Proof of Theorem 4
Theorem 4 is proved by showing that mappings f and g send feasible points between the two SDP formulations and are inverse identity maps.
- Mapping f: Mapping f sends every point in FBIM-SDP to a point in FBFM-SDP.The proof checks the target point against constraints (10a)–(10f).
- Mapping g: Mapping g sends every point in FBFM-SDP to a point in FBIM-SDP.Constraint (5f) is established using (15), while the remaining constraints are checked directly.
- Inverse mappings: The proof verifies that f ◦ g and g ◦ f are identity maps, completing Theorem 4.Thus the two mappings undo each other on their respective feasible sets.
- Follow-up verification: For points mapped by f, the proof reduces the remaining claim to verifying the relevant condition before completing Theorem 5.The cited passage introduces this verification for (s, v, W) ∈ FBIM-SDP.